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 Subject: "Relatively prime numbers" Previous Topic | Next Topic
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Ginger guest
Nov-25-01, 08:57 PM (EST)

"Relatively prime numbers"

 Hello - I have a problem. I am a student in elementary eduducation and I have come across a problem that I don't understand. Please help if you can. Two integers, m and n are said to be relatively prime if their greatest common divisor is 1, that is GCD(m,n) = 1. 0/ means 0 with a slash through it, my keyboard wouldn't do it.If m is a positive integer, 0/(m) denote the number of positive integers less than or equal to m that are also relatively prime to m.1. Calculate 0/(2), 0/(3), 0/(5) and 0/(7).I think once I understand what this means, I will be able to move forward with the problem.Thank you.

alexb
Charter Member
672 posts
Nov-25-01, 09:12 PM (EST)    1. "RE: Relatively prime numbers"
In response to message #0

 >Two integers, m and n are said to be relatively prime if >their greatest common divisor is 1, that is GCD(m,n) = 1. The terminology varies. Alternatively, two integers with no common divisors save for 1 are called mututally prime or coprime. (By definition, 1 is relatively prime with any integer.)> 0/ means 0 with a slash through it, my keyboard wouldn't do >it. This is in fact the Greek letter phi - f. f(n) denotes Euler's phi function or Euler's totient function. (This is in case you would like - which would make sense too - to search this site or the Web for information about that function and its properties.)>If m is a positive integer, 0/(m) denote the number of >positive integers less than or equal to m that are also >relatively prime to m. >>1. Calculate 0/(2), 0/(3), 0/(5) and 0/(7). f(2) = 1There is just one positive integer less than 2. It's 1, which is relatively prime to 2.f(3) = 2There are just two positive integers less than 3: 1 and 2. Both are relatively prime to 3.f(5) = 4All of 1, 2, 3, 4 are relatively prime to 5.f(4) = 2Of 1, 2, 3 the first and the last are relatively prime to 4.>I think once I understand what this means, I will be able to >move forward with the problem. Hope the above helps.

Natalie guest
Nov-26-01, 08:50 PM (EST)

2. "RE: Relatively prime numbers"
In response to message #0

 I think that the problem only wants you to come up with a number that ALONG WITH the one given, makes the pair relatively prime.I tell my middle school students that the easiest way to do this is to choose any other prime number.For example, in your problem 0/(your notation) 0/2 would be asking you to find another number, along with 2, that makes them relatively prime. No even number will work, but any odd number would work, especially any prime. Same for 3 and 7, just find a number that doesn't have 3 as a factor, or 7 as a factor and those numbers are relatively prime to 3 and 7.Good luck!Natalie alexb
Charter Member
672 posts
Nov-26-01, 08:54 PM (EST)    3. "RE: Relatively prime numbers"
In response to message #2

 >I think that the problem only wants you to come up with a >number that ALONG WITH the one given, makes the pair >relatively prime. Do not see what caused you to think that. The original poster tated clearly thatIf m is a positive integer, 0/(m) denote the number of positive integers less than or equal to m that are also relatively prime to m.>I tell my middle school students that the easiest way to do >this is to choose any other prime number. What other prime number is less than 2?>For example, in your problem 0/(your notation) 0/2 would >be asking you to find another number, along with 2, that >makes them relatively prime. No even number will work, but >any odd number would work, especially any prime. Again, no. Because you have to select among numbers less than 2!

Ginger guest
Nov-30-01, 06:32 PM (EST)

4. "RE: Relatively prime numbers"
In response to message #0

 Thank you very much for the help. After much looking, I also found what you were saying. As yes, you are right. Do you have any conjectures when it comes to the non-prime numbers? I thought I saw a pattern, but then it doesn't always work. I do see that the phi of the factors of (m) add up to (m), in some cases. Which is interesting, however, I do not see that there is a standard rule for non primes. What do you think?Many ThanksGinger alexb
Charter Member
672 posts
Nov-30-01, 06:41 PM (EST)    5. "RE: Relatively prime numbers"
In response to message #4

 >Do >you have any conjectures when it comes to the non-prime >numbers? Euler's function is easily defined for any integer. Please have a look at https://www.cut-the-knot.com/blue/Euler.shtml

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